Problem 2 , River pollution

You want to provide decision makers with the best estimate of the pollutant concentration in a river at different downstream distances to give them info on which to base their decisions for water quality management. You were asked to adopt the Streeter-Phelps model to approximate concentrations of the chemical in the river.The average flow velocity is 0.305 m/s. At the source, the BOD concentration is 10.0 mg/L and the dissolved oxygen deficit is 0.0 mg/L.From field sampling, de-oxygenation (kd) and re-aeration (ka) rate constants are estimated to be 0.6 d-1 and 2.0 d-1, respectively.

• Using Excel (or any other software of choice),
• calculate the BOD concentration and the dissolved oxygen deficit at downstream distances up to 200 km (use 1 km space intervals).
• Plot the BOD concentration and the dissolved oxygen deficit as a function of downstream distance. Add your plot below (make sure you add labels/legends and a caption).
• Find the critical distance, the critical deficit, and the time at which the minimum D.O. occurs. You can either solve this analytically or using Excel/Matlab/Python/R/etc., but please show the procedure you followed (and not just the answer).
• Discuss your findings (i.e., results and plots from points 2 and 3 above).

Problem #3: Mathematics of Growth

Suppose that the population of grizzly bears in a national park grows according to the following logistic differential equation:

dN/ dt=5N(1-N/2500)

N is the number of bears at time t in years.

Using Excel (or Matlab, Pyhton, R, IDL, …):

Find how the population grows with time if the initial population is 100 bears, and sketch a graph of N(t). Make sure you add labels/legends/captions to your plot. Copy the figure you produced here

• How does your solution change if N(0) = 1500? (Add the figure below)
• If N(0) = 3000? (Add the figure below)
• How many bears are in the park after 3 years for each case? Discuss your results.